Convert Cartesian to Polar Coordinates

Point in the Polar and Cartesian System

To convert a point from the cartesian coordinate system to the polar coordinate system the pythagorean theorem and the arctangent function are used to calculate the radius and angle of the correponding point.

Convert Cartesian to Polar Example

For example, to convert the cartesian point to polar coordinates. First we solve for the hypotenuse length of the right triangle to find the length of the radius:

Then use the arctangent function to find the angle:

The cartesian point is equivalent to the polar point . Note, if the same calculation is performed with a calculator set to degrees instead of radians the point would be . The right triangle formed by both points is shown below:

Point in the Polar and Cartesian System

Explanation

Both systems describe the position of a point in space. A point in the Polar Coordinate System is defined in terms of a radius and an angle: . A point in the Cartesian Coordinate System is defined in terms of a and component: . Both define the point relative to the origin of the system. Geometrically, the two points can be described by the right triangle below.

Polar Coordinates Point

The pythagorean theorem relates the squared sides together on a right-triangle. Since the component corresponds to the adjacent side of the right triangle and the component corresponds to the opposite side, the equation can be rearranged to give the length of the hypotenuse which corresponds to the length of the radius in a polar coordinate.

The arctangent function returns the angle of a right-triangle given the ratio of its opposite side over its adjacent side.

Good Links

Cartesian Coordinate System | Concept

The Cartesian Coordinate System describes space of one, two, and three dimensions. Each point in space is represented by its distance relative to the origin of the system.

Polar Coordinate System | Concept

The Polar Coordinate System describes points in space using an angle and radius relative to the origin.

Radians | Concept

Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

Cosine Function | Function

The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the right-triangle's adjacent side over its hypotenuse.

Sine Function | Function

The sine function returns the sine of a number provided in radians. In geometric terms, given the angle of a right-triangle as input, the function returns the ratio of the triangle's opposite side over its hypotenuse.

Unit Circle | Concept

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.